Responsive uses a library such as Bootstrap to detect browser size and capabilities on the client and select which pre-programmed layout to use. It's quite popular due to Bootstrap.

Adaptive tells the server what the browser size and capabilities are, and sends this information to the server, which serves up a customized page specific to those browser specs. Nothing extra is done on the client side. This doesn't seem to be as popular as Responsive.

Most of the buzz these days seems to be Responsive Web Design, so you will probably find the largest community of developers and most up to date support libraries. It seems to work well, at the cost of a little extra JS and CSS having to be sent over, and has the advantage that the correct page sizing etc. is immediate (i.e., you don't have to load a general-purpose page to send browser specs back to the server).